What is Quantum Computing?

Kevin Yuh

June 28, 2012

DefinitionsQubit The unit of information in quantum computing. Defined as a superposition of two states |0〉 and |1〉,

and represented as |ψ〉 = a |0〉 + b |1〉, a, b ∈ C, where |a|2 + |b|2 = 1. State can also be represented as

the matrix[ab

]. Physically, |a|2 and |b|2 represent the probability that, upon measurement, 0 or 1

will be obtained, respectively.

N-qubit system A superposition of 2n states, represented as |ψ〉 =∑2n−1i=0 ci |xi〉, ci ∈ C, where

∑2n−1i=0 ci =

1 and xi is the i-th bitstring (0...0, 0...1, ..., 1...1) of size n. Conveniently represented as

c0c1...cn

.Quantum gate Mathematically, a 2n×2n linear map U from one n-bit quantum state to the next. Produces

a new quantum state |ψ′〉 = U |ψ〉 = U

c0c1...cn

=

c′0c′1...c′n

. U must ensure that∑2n−1i=0 c′i = 1, i.e. U

must be a unitary transformation.

Hadamard gate One of the universal quantum operations, defined by H = 1√2

[1 11 −1

]. Important

because it transforms any pure state into a superposition of its possible states.

Phase gate Another universal quantum operation, defined by Φφ =

[1 00 eiφ

].

Controlled-NOT (CNOT) gate A universal quantum operation on two qubits simultaneously, defined by

UCNOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

.Bit-flip gate Defined by X =

[0 11 0

]. Flips the coefficients of |0〉 and |1〉 .

Phase-flip gate Defined by Z =

[1 00 −1

]. Flips b |1〉 to −b |1〉.

References[1] David Deutsch and Richard Jozsa. Rapid solutions of problems by quantum computation. In Proceedings

of the Royal Society of London, 1992.1

[2] Dieter Heiss. Fundamentals of Quantum Information. Springer, Berlin, Germany, 2002.

[3] Tzvetan S. Metodi and Frederic T. Chong. Quantum Computing for Computer Architects. Morgan andClaypool Publishers, USA, 2006.

[4] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. CambridgeUniversity Press, United Kingdom, 2000.

[5] Marek Perkowski. Quantum logic. Non-physics introduction to quantum computing.

2